$d_p$ convergence and $\epsilon$-regularity theorems for entropy and scalar curvature lower bounds

TL;DR

This paper introduces a new convergence notion called $d_p$ convergence for Riemannian manifolds with near-nonnegative scalar curvature and entropy bounds, establishing regularity and compactness results in this framework.

Contribution

It develops the concept of $d_p$ convergence to analyze limits of manifolds with scalar curvature and entropy bounds, overcoming limitations of classical convergence notions.

Findings

Sequences with small scalar curvature and entropy bounds converge in $d_p$ to rectifiable Riemannian spaces.

Under $d_p$ convergence, such spaces are close to Euclidean space, leading to an $ ext{epsilon}$-regularity theorem.

The framework allows for applications like $L^ ext{infinity}$-Sobolev embeddings and $L^p$ scalar curvature bounds for $p<1$.

Abstract

Consider a sequence of Riemannian manifolds $(M^n_i,g_i)$ with scalar curvatures and entropies bounded below by small constants $R_i,\mu_i \geq-\epsilon_i$. The goal of this paper is to understand notions of convergence and the structure of limits for such spaces. Even in the seemingly rigid case $\epsilon_i\to 0$, we construct examples showing that such a sequence may converge wildly in the Gromov-Hausdorff or Intrinsic Flat sense. On the other hand, we will see that these classical notions of convergence are the incorrect ones to consider. Indeed, even a metric space is the wrong underlying category to be working on. Instead, we introduce $d_p$ convergence, a weaker notion of convergence that is valid for a class of rectifiable Riemannian spaces. These rectifiable spaces have well-behaved topology, measure theory, and analysis, though potentially there will be no reasonably associated distance function. Under the $d_p$ notion of closeness, a space with almost nonnegative scalar curvature and small entropy bounds must in fact be close to Euclidean space; this will constitute our $\epsilon$-regularity theorem. More generally, we have a compactness theorem saying that sequences of Riemannian manifolds $(M^n_i,g_i)$ with small lower scalar curvature and entropy bounds $R_i,\mu_i \geq -\epsilon$ must $d_p$ converge to such a rectifiable Riemannian space $X$. Comparing to the first paragraph, the distance functions of $M_i$ may be degenerating, even though in a well-defined sense the analysis cannot be. Applications for manifolds with small scalar and entropy lower bounds include an $L^\infty$-Sobolev embedding and apriori $L^p$ scalar curvature bounds for $p<1$.